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                <h2 id="「力扣」第-64-题：最小路径和（中等）"><a href="#「力扣」第-64-题：最小路径和（中等）" class="headerlink" title="「力扣」第 64 题：最小路径和（中等）"></a>「力扣」第 64 题：最小路径和（中等）</h2><blockquote>
<p>很简单的「动态规划」问题，熟悉「自底向上」求解问题的思路。</p>
</blockquote>
<ul>
<li><a href="https://leetcode-cn.com/problems/minimum-path-sum/description/" target="_blank" rel="noopener">链接</a></li>
</ul>
<blockquote>
<p>给定一个包含非负整数的 <em>m</em> x <em>n</em> 网格，请找出一条从左上角到右下角的路径，使得路径上的数字总和为最小。</p>
<p><strong>说明</strong>：每次只能向下或者向右移动一步。</p>
<p><strong>示例</strong>：</p>
<pre><code>输入:
[
  [1, 3, 1],
  [1, 5, 1],
  [4, 2, 1]
]
输出: 7
解释: 因为路径 1 → 3 → 1 → 1 → 1 的总和最小。</code></pre></blockquote>
<h3 id="方法：动态规划"><a href="#方法：动态规划" class="headerlink" title="方法：动态规划"></a>方法：动态规划</h3><p>每一步只能左移或者下移。题目中给出了 <code>grid</code> 非负整数，可以保证走得越长，sum 的值越大；只能向右走或者向下走，保证了在非负整数矩阵的情况下，sum 的最小值存在。在分析清楚以后，我们可以直接在 grid 矩阵上原地修改，直接给出动态规划的解法，十分简单，逻辑也很清晰，核心代码不超过 10 行。</p>
<p>状态很好定义，题目中问什么，状态就定义成什么：<code>dp[i][j]</code>。</p>
<p>Java 代码：</p>
<pre class="line-numbers language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>

        <span class="token comment" spellcheck="true">// 直接在 grid 数组上操作</span>

    <span class="token keyword">public</span> <span class="token keyword">int</span> <span class="token function">minPathSum</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token punctuation">]</span> grid<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>grid <span class="token operator">==</span> null<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">int</span> m <span class="token operator">=</span> grid<span class="token punctuation">.</span>length<span class="token punctuation">;</span>
        <span class="token keyword">int</span> n <span class="token operator">=</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length<span class="token punctuation">;</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> n<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">=</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">+</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span>i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> m<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">=</span> grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">+</span> grid<span class="token punctuation">[</span>i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> m<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">&lt;</span> n<span class="token punctuation">;</span> j<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> Math<span class="token punctuation">.</span><span class="token function">min</span><span class="token punctuation">(</span>grid<span class="token punctuation">[</span>i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">,</span> grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">)</span> <span class="token operator">+</span> grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> grid<span class="token punctuation">[</span>m <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span>n <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">main</span><span class="token punctuation">(</span>String<span class="token punctuation">[</span><span class="token punctuation">]</span> args<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token punctuation">]</span> grid <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token punctuation">{</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">5</span><span class="token punctuation">}</span><span class="token punctuation">,</span> <span class="token punctuation">{</span><span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">}</span><span class="token punctuation">;</span>
        Solution solution <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">Solution</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> minPathSum <span class="token operator">=</span> solution<span class="token punctuation">.</span><span class="token function">minPathSum</span><span class="token punctuation">(</span>grid<span class="token punctuation">)</span><span class="token punctuation">;</span>
        System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>minPathSum<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<p>Python 代码：</p>
<pre class="line-numbers language-python"><code class="language-python"><span class="token keyword">from</span> typing <span class="token keyword">import</span> List


<span class="token keyword">class</span> <span class="token class-name">Solution</span><span class="token punctuation">:</span>
    <span class="token keyword">def</span> <span class="token function">minPathSum</span><span class="token punctuation">(</span>self<span class="token punctuation">,</span> grid<span class="token punctuation">:</span> List<span class="token punctuation">[</span>List<span class="token punctuation">[</span>int<span class="token punctuation">]</span><span class="token punctuation">]</span><span class="token punctuation">)</span> <span class="token operator">-</span><span class="token operator">></span> int<span class="token punctuation">:</span>

        m <span class="token operator">=</span> len<span class="token punctuation">(</span>grid<span class="token punctuation">)</span>
        <span class="token keyword">if</span> m <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">:</span>
            <span class="token keyword">return</span> <span class="token number">0</span>
        n <span class="token operator">=</span> len<span class="token punctuation">(</span>grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">)</span>

        <span class="token keyword">for</span> col <span class="token keyword">in</span> range<span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">,</span> n<span class="token punctuation">)</span><span class="token punctuation">:</span>
            <span class="token comment" spellcheck="true"># 第 0 行特殊处理，不要忘记了</span>
            grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span>col<span class="token punctuation">]</span> <span class="token operator">+=</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span>col <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span>
        <span class="token keyword">for</span> row <span class="token keyword">in</span> range<span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">,</span> m<span class="token punctuation">)</span><span class="token punctuation">:</span>
            grid<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">+=</span> grid<span class="token punctuation">[</span>row <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span>
            <span class="token keyword">for</span> col <span class="token keyword">in</span> range<span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">,</span> n<span class="token punctuation">)</span><span class="token punctuation">:</span>
                grid<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">[</span>col<span class="token punctuation">]</span> <span class="token operator">+=</span> min<span class="token punctuation">(</span>grid<span class="token punctuation">[</span>row <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span>col<span class="token punctuation">]</span><span class="token punctuation">,</span> grid<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">[</span>col <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">)</span>
        <span class="token keyword">return</span> grid<span class="token punctuation">[</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span>


<span class="token keyword">if</span> __name__ <span class="token operator">==</span> <span class="token string">'__main__'</span><span class="token punctuation">:</span>
    grid <span class="token operator">=</span> <span class="token punctuation">[</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">,</span>
            <span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">5</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">,</span>
            <span class="token punctuation">[</span><span class="token number">4</span><span class="token punctuation">,</span> <span class="token number">2</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">]</span>

    s <span class="token operator">=</span> Solution<span class="token punctuation">(</span><span class="token punctuation">)</span>
    res <span class="token operator">=</span> s<span class="token punctuation">.</span>minPathSum<span class="token punctuation">(</span>grid<span class="token punctuation">)</span>
    <span class="token keyword">print</span><span class="token punctuation">(</span>res<span class="token punctuation">)</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<ul>
<li>技巧：表格复用</li>
</ul>
<p>Java 代码：</p>
<pre class="line-numbers language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">Solution</span> <span class="token punctuation">{</span>

    <span class="token keyword">public</span> <span class="token keyword">int</span> <span class="token function">minPathSum</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token punctuation">]</span> grid<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>grid <span class="token operator">==</span> null<span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">int</span> m <span class="token operator">=</span> grid<span class="token punctuation">.</span>length<span class="token punctuation">;</span>
        <span class="token keyword">int</span> n <span class="token operator">=</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">.</span>length<span class="token punctuation">;</span>

        <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> dp <span class="token operator">=</span> <span class="token keyword">new</span> <span class="token class-name">int</span><span class="token punctuation">[</span>n<span class="token punctuation">]</span><span class="token punctuation">;</span>
        dp<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">=</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">;</span>

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> m<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> j <span class="token operator">&lt;</span> n<span class="token punctuation">;</span> j<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">if</span> <span class="token punctuation">(</span>i <span class="token operator">==</span> <span class="token number">0</span> <span class="token operator">&amp;&amp;</span> j <span class="token operator">></span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                    dp<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> grid<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">+</span> dp<span class="token punctuation">[</span>j <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
                <span class="token punctuation">}</span> <span class="token keyword">else</span> <span class="token keyword">if</span> <span class="token punctuation">(</span>j <span class="token operator">==</span> <span class="token number">0</span> <span class="token operator">&amp;&amp;</span> i <span class="token operator">></span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                    dp<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> dp<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">+</span> grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
                <span class="token punctuation">}</span> <span class="token keyword">else</span> <span class="token keyword">if</span> <span class="token punctuation">(</span>i <span class="token operator">!=</span> <span class="token number">0</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                    dp<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> Math<span class="token punctuation">.</span><span class="token function">min</span><span class="token punctuation">(</span>dp<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">,</span> dp<span class="token punctuation">[</span>j <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">)</span> <span class="token operator">+</span> grid<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
                <span class="token punctuation">}</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">return</span> dp<span class="token punctuation">[</span>n <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<p><strong>复杂度分析</strong>：</p>
<ul>
<li>时间复杂度：<img src="data:image/svg+xml,%3Csvg%20xmlns%3Axlink%3D%22http%3A%2F%2Fwww.w3.org%2F1999%2Fxlink%22%20width%3D%228.088ex%22%20height%3D%222.843ex%22%20style%3D%22vertical-align%3A%20-0.838ex%3B%22%20viewBox%3D%220%20-863.1%203482.5%201223.9%22%20role%3D%22img%22%20focusable%3D%22false%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20aria-labelledby%3D%22MathJax-SVG-1-Title%22%3E%0A%3Ctitle%20id%3D%22MathJax-SVG-1-Title%22%3EO(MN)%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4F%22%20d%3D%22M740%20435Q740%20320%20676%20213T511%2042T304%20-22Q207%20-22%20138%2035T51%20201Q50%20209%2050%20244Q50%20346%2098%20438T227%20601Q351%20704%20476%20704Q514%20704%20524%20703Q621%20689%20680%20617T740%20435ZM637%20476Q637%20565%20591%20615T476%20665Q396%20665%20322%20605Q242%20542%20200%20428T157%20216Q157%20126%20200%2073T314%2019Q404%2019%20485%2098T608%20313Q637%20408%20637%20476Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-28%22%20d%3D%22M94%20250Q94%20319%20104%20381T127%20488T164%20576T202%20643T244%20695T277%20729T302%20750H315H319Q333%20750%20333%20741Q333%20738%20316%20720T275%20667T226%20581T184%20443T167%20250T184%2058T225%20-81T274%20-167T316%20-220T333%20-241Q333%20-250%20318%20-250H315H302L274%20-226Q180%20-141%20137%20-14T94%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4D%22%20d%3D%22M289%20629Q289%20635%20232%20637Q208%20637%20201%20638T194%20648Q194%20649%20196%20659Q197%20662%20198%20666T199%20671T201%20676T203%20679T207%20681T212%20683T220%20683T232%20684Q238%20684%20262%20684T307%20683Q386%20683%20398%20683T414%20678Q415%20674%20451%20396L487%20117L510%20154Q534%20190%20574%20254T662%20394Q837%20673%20839%20675Q840%20676%20842%20678T846%20681L852%20683H948Q965%20683%20988%20683T1017%20684Q1051%20684%201051%20673Q1051%20668%201048%20656T1045%20643Q1041%20637%201008%20637Q968%20636%20957%20634T939%20623Q936%20618%20867%20340T797%2059Q797%2055%20798%2054T805%2050T822%2048T855%2046H886Q892%2037%20892%2035Q892%2019%20885%205Q880%200%20869%200Q864%200%20828%201T736%202Q675%202%20644%202T609%201Q592%201%20592%2011Q592%2013%20594%2025Q598%2041%20602%2043T625%2046Q652%2046%20685%2049Q699%2052%20704%2061Q706%2065%20742%20207T813%20490T848%20631L654%20322Q458%2010%20453%205Q451%204%20449%203Q444%200%20433%200Q418%200%20415%207Q413%2011%20374%20317L335%20624L267%20354Q200%2088%20200%2079Q206%2046%20272%2046H282Q288%2041%20289%2037T286%2019Q282%203%20278%201Q274%200%20267%200Q265%200%20255%200T221%201T157%202Q127%202%2095%201T58%200Q43%200%2039%202T35%2011Q35%2013%2038%2025T43%2040Q45%2046%2065%2046Q135%2046%20154%2086Q158%2092%20223%20354T289%20629Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4E%22%20d%3D%22M234%20637Q231%20637%20226%20637Q201%20637%20196%20638T191%20649Q191%20676%20202%20682Q204%20683%20299%20683Q376%20683%20387%20683T401%20677Q612%20181%20616%20168L670%20381Q723%20592%20723%20606Q723%20633%20659%20637Q635%20637%20635%20648Q635%20650%20637%20660Q641%20676%20643%20679T653%20683Q656%20683%20684%20682T767%20680Q817%20680%20843%20681T873%20682Q888%20682%20888%20672Q888%20650%20880%20642Q878%20637%20858%20637Q787%20633%20769%20597L620%207Q618%200%20599%200Q585%200%20582%202Q579%205%20453%20305L326%20604L261%20344Q196%2088%20196%2079Q201%2046%20268%2046H278Q284%2041%20284%2038T282%2019Q278%206%20272%200H259Q228%202%20151%202Q123%202%20100%202T63%202T46%201Q31%201%2031%2010Q31%2014%2034%2026T39%2040Q41%2046%2062%2046Q130%2049%20150%2085Q154%2091%20221%20362L289%20634Q287%20635%20234%20637Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-29%22%20d%3D%22M60%20749L64%20750Q69%20750%2074%20750H86L114%20726Q208%20641%20251%20514T294%20250Q294%20182%20284%20119T261%2012T224%20-76T186%20-143T145%20-194T113%20-227T90%20-246Q87%20-249%2086%20-250H74Q66%20-250%2063%20-250T58%20-247T55%20-238Q56%20-237%2066%20-225Q221%20-64%20221%20250T66%20725Q56%20737%2055%20738Q55%20746%2060%20749Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4F%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%22763%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4D%22%20x%3D%221153%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4E%22%20x%3D%222204%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%223093%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E" alt="img">，这里 <img src="data:image/svg+xml,%3Csvg%20xmlns%3Axlink%3D%22http%3A%2F%2Fwww.w3.org%2F1999%2Fxlink%22%20width%3D%222.442ex%22%20height%3D%222.176ex%22%20style%3D%22vertical-align%3A%20-0.338ex%3B%22%20viewBox%3D%220%20-791.3%201051.5%20936.9%22%20role%3D%22img%22%20focusable%3D%22false%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20aria-labelledby%3D%22MathJax-SVG-1-Title%22%3E%0A%3Ctitle%20id%3D%22MathJax-SVG-1-Title%22%3EM%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4D%22%20d%3D%22M289%20629Q289%20635%20232%20637Q208%20637%20201%20638T194%20648Q194%20649%20196%20659Q197%20662%20198%20666T199%20671T201%20676T203%20679T207%20681T212%20683T220%20683T232%20684Q238%20684%20262%20684T307%20683Q386%20683%20398%20683T414%20678Q415%20674%20451%20396L487%20117L510%20154Q534%20190%20574%20254T662%20394Q837%20673%20839%20675Q840%20676%20842%20678T846%20681L852%20683H948Q965%20683%20988%20683T1017%20684Q1051%20684%201051%20673Q1051%20668%201048%20656T1045%20643Q1041%20637%201008%20637Q968%20636%20957%20634T939%20623Q936%20618%20867%20340T797%2059Q797%2055%20798%2054T805%2050T822%2048T855%2046H886Q892%2037%20892%2035Q892%2019%20885%205Q880%200%20869%200Q864%200%20828%201T736%202Q675%202%20644%202T609%201Q592%201%20592%2011Q592%2013%20594%2025Q598%2041%20602%2043T625%2046Q652%2046%20685%2049Q699%2052%20704%2061Q706%2065%20742%20207T813%20490T848%20631L654%20322Q458%2010%20453%205Q451%204%20449%203Q444%200%20433%200Q418%200%20415%207Q413%2011%20374%20317L335%20624L267%20354Q200%2088%20200%2079Q206%2046%20272%2046H282Q288%2041%20289%2037T286%2019Q282%203%20278%201Q274%200%20267%200Q265%200%20255%200T221%201T157%202Q127%202%2095%201T58%200Q43%200%2039%202T35%2011Q35%2013%2038%2025T43%2040Q45%2046%2065%2046Q135%2046%20154%2086Q158%2092%20223%20354T289%20629Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4D%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E" alt="img"> 和 <img src="data:image/svg+xml,%3Csvg%20xmlns%3Axlink%3D%22http%3A%2F%2Fwww.w3.org%2F1999%2Fxlink%22%20width%3D%222.064ex%22%20height%3D%222.176ex%22%20style%3D%22vertical-align%3A%20-0.338ex%3B%22%20viewBox%3D%220%20-791.3%20888.5%20936.9%22%20role%3D%22img%22%20focusable%3D%22false%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20aria-labelledby%3D%22MathJax-SVG-1-Title%22%3E%0A%3Ctitle%20id%3D%22MathJax-SVG-1-Title%22%3EN%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4E%22%20d%3D%22M234%20637Q231%20637%20226%20637Q201%20637%20196%20638T191%20649Q191%20676%20202%20682Q204%20683%20299%20683Q376%20683%20387%20683T401%20677Q612%20181%20616%20168L670%20381Q723%20592%20723%20606Q723%20633%20659%20637Q635%20637%20635%20648Q635%20650%20637%20660Q641%20676%20643%20679T653%20683Q656%20683%20684%20682T767%20680Q817%20680%20843%20681T873%20682Q888%20682%20888%20672Q888%20650%20880%20642Q878%20637%20858%20637Q787%20633%20769%20597L620%207Q618%200%20599%200Q585%200%20582%202Q579%205%20453%20305L326%20604L261%20344Q196%2088%20196%2079Q201%2046%20268%2046H278Q284%2041%20284%2038T282%2019Q278%206%20272%200H259Q228%202%20151%202Q123%202%20100%202T63%202T46%201Q31%201%2031%2010Q31%2014%2034%2026T39%2040Q41%2046%2062%2046Q130%2049%20150%2085Q154%2091%20221%20362L289%20634Q287%20635%20234%20637Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4E%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E" alt="img"> 分别是矩阵的长和宽；</li>
<li>空间复杂度：<img src="data:image/svg+xml,%3Csvg%20xmlns%3Axlink%3D%22http%3A%2F%2Fwww.w3.org%2F1999%2Fxlink%22%20width%3D%225.646ex%22%20height%3D%222.843ex%22%20style%3D%22vertical-align%3A%20-0.838ex%3B%22%20viewBox%3D%220%20-863.1%202431%201223.9%22%20role%3D%22img%22%20focusable%3D%22false%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20aria-labelledby%3D%22MathJax-SVG-1-Title%22%3E%0A%3Ctitle%20id%3D%22MathJax-SVG-1-Title%22%3EO(N)%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4F%22%20d%3D%22M740%20435Q740%20320%20676%20213T511%2042T304%20-22Q207%20-22%20138%2035T51%20201Q50%20209%2050%20244Q50%20346%2098%20438T227%20601Q351%20704%20476%20704Q514%20704%20524%20703Q621%20689%20680%20617T740%20435ZM637%20476Q637%20565%20591%20615T476%20665Q396%20665%20322%20605Q242%20542%20200%20428T157%20216Q157%20126%20200%2073T314%2019Q404%2019%20485%2098T608%20313Q637%20408%20637%20476Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-28%22%20d%3D%22M94%20250Q94%20319%20104%20381T127%20488T164%20576T202%20643T244%20695T277%20729T302%20750H315H319Q333%20750%20333%20741Q333%20738%20316%20720T275%20667T226%20581T184%20443T167%20250T184%2058T225%20-81T274%20-167T316%20-220T333%20-241Q333%20-250%20318%20-250H315H302L274%20-226Q180%20-141%20137%20-14T94%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-4E%22%20d%3D%22M234%20637Q231%20637%20226%20637Q201%20637%20196%20638T191%20649Q191%20676%20202%20682Q204%20683%20299%20683Q376%20683%20387%20683T401%20677Q612%20181%20616%20168L670%20381Q723%20592%20723%20606Q723%20633%20659%20637Q635%20637%20635%20648Q635%20650%20637%20660Q641%20676%20643%20679T653%20683Q656%20683%20684%20682T767%20680Q817%20680%20843%20681T873%20682Q888%20682%20888%20672Q888%20650%20880%20642Q878%20637%20858%20637Q787%20633%20769%20597L620%207Q618%200%20599%200Q585%200%20582%202Q579%205%20453%20305L326%20604L261%20344Q196%2088%20196%2079Q201%2046%20268%2046H278Q284%2041%20284%2038T282%2019Q278%206%20272%200H259Q228%202%20151%202Q123%202%20100%202T63%202T46%201Q31%201%2031%2010Q31%2014%2034%2026T39%2040Q41%2046%2062%2046Q130%2049%20150%2085Q154%2091%20221%20362L289%20634Q287%20635%20234%20637Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-29%22%20d%3D%22M60%20749L64%20750Q69%20750%2074%20750H86L114%20726Q208%20641%20251%20514T294%20250Q294%20182%20284%20119T261%2012T224%20-76T186%20-143T145%20-194T113%20-227T90%20-246Q87%20-249%2086%20-250H74Q66%20-250%2063%20-250T58%20-247T55%20-238Q56%20-237%2066%20-225Q221%20-64%20221%20250T66%20725Q56%20737%2055%20738Q55%20746%2060%20749Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4F%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%22763%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-4E%22%20x%3D%221153%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%222041%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E" alt="img">。</li>
</ul>
<p>（本节完）</p>

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                            「力扣」第 70 题：爬楼梯
斐波拉契数列，画出树形结构，发现大量重叠子问题。「动态规划」告诉我们「自顶向上」求解问题的思路。


链接
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假设你正在爬楼梯。需要 n 阶你才能到达楼顶。
每次你可以爬 1 或 2 个台阶。你有
                        
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                                    专题 15：动态规划
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                            「力扣」第 63 题：不同路径 II（中等）
这题是动态规划的经典问题：理解「无后效性」。
1、状态压缩：具体讲解一下，状态行是如何压缩的；
2、哨兵技巧，回避边界条件的讨论；
3、数学方法；
4、只能向下和只能向右，说明符合动态规划的无后
                        
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                                    专题 15：动态规划
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